On the multiplication operators on spaces of analytic functions

We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from $H^{∞}(Ω)$. Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators.